CSE 326: Data Structures Dijkstra s Algorithm. James Fogarty Autumn 2007

Transcription

1 SE 6: Data Structures Dijkstra s lgorithm James Fogarty utumn 007

2 Dijkstra, Edsger Wybe Legendary figure in computer science; was a professor at University of Texas. Supported teaching introductory computer courses without computers (pencil and paper programming) Supposedly wouldn t (until very late in life) read his ; so, his staff had to print out messages and put them in his box. E.W. Dijkstra (90-00) 97 Turning ward Winner, Programming Languages, semaphores, and

3 Dijkstra s lgorithm: Idea dapt FS to handle weighted graphs Two kinds of vertices: Finished or known vertices Shortest distance has been computed Unknown vertices Have tentative distance

4 Dijkstra s lgorithm: Idea t each step: ) Pick closest unknown vertex ) dd it to known vertices ) Update distances

5 Dijkstra s lgorithm: Pseudocode Initialize the cost of each node to Initialize the cost of the source to 0 While there are unknown nodes left in the graph Select an unknown node b with the lowest cost Mark b as known For each node a adjacent to b a s cost = min(a s old cost, b s cost + cost of (b, a)) a s prev path node = b 5

6 Important Features Once a vertex is made known, the cost of the shortest path to that node is known While a vertex is still not known, another shorter path to it might still be found The shortest path itself can found by following the backward pointers stored in node.path 6

7 Dijkstra s lgorithm in action D 0 F H E G Vertex Visited? ost Found by 0???? D?? E?? F?? G?? H?? 7

8 Dijkstra s lgorithm in action D 0 F H E G Vertex Visited? ost Found by Y 0 <= <= D <= E?? F?? G?? H?? 8

9 Dijkstra s lgorithm in action D 0 F H E G Vertex Visited? ost Found by Y 0 <= Y D <= E <= F?? G?? H?? 9

10 Dijkstra s lgorithm in action D 0 F H E G Vertex Visited? ost Found by Y 0 Y Y D <= E <= F <= G?? H?? 0

11 Dijkstra s lgorithm in action D 0 F H E G Vertex Visited? ost Found by Y 0 Y Y D Y E <= F <= G?? H??

12 Dijkstra s lgorithm in action D 0 7 F H E G Vertex Visited? ost Found by Y 0 Y Y D Y E <= F Y G?? H <=7 F

13 Dijkstra s lgorithm in action D 0 7 F H E G Vertex Visited? ost Found by Y 0 Y Y D Y E <= F Y G <=8 H Y 7 H F 8

14 Dijkstra s lgorithm in action D 0 7 F H E G Vertex Visited? ost Found by Y 0 Y Y D Y E <= F Y G Y 8 H Y 7 G H F 8

15 Dijkstra s lgorithm in action D 0 7 F H E G Vertex Visited? ost Found by Y 0 Y Y D Y E Y F Y G Y 8 H Y 7 G H F 8 5

16 Your turn s v 0 v 5 v v v V Visited? ost Found by 6 5 v0 v v 5 0 v 6 v v v v5 v6 6

17 Dijkstra s lg: Implementation Initialize the cost of each node to Initialize the cost of the source to 0 While there are unknown nodes left in the graph Select the unknown node b with the lowest cost Mark b as known For each node a adjacent to b a s cost = min(a s old cost, b s cost + cost of (b, a)) a s prev path node = b (if we updated a s cost) What data structures should we use? Running time? 7

18 void Graph::dijkstra(Vertex s){ Vertex v,w; Initialize s.dist = 0 and set dist of all other vertices to infinity Sounds like adjacency lists } while (there exist unknown vertices, find the one b with the smallest distance) b.known = true; } for each a adjacent to b if (!a.known) if (b.dist + weight(b,a) < a.dist){ a.dist = (b.dist + weight(b,a)); a.path = b; } Sounds like deletemin on a heap Sounds like decreasekey Running time: O( E log V ) there are E edges to examine, and each one causes a heap operation of time O(log V ) 8

19 Dijkstra s lgorithm: Summary lassic algorithm for solving SSSP in weighted graphs without negative weights greedy algorithm (irrevocably makes decisions without considering future consequences) Intuition for correctness: shortest path from source vertex to itself is 0 cost of going to adjacent nodes is at most edge weights cheapest of these must be shortest path to that node update paths for new node and continue picking cheapest path 9

20 orrectness: The loud Proof V Next shortest path from inside the known cloud etter path to V? No! W The Known loud Source How does Dijkstra s decide which vertex to add to the Known set next? If path to V is shortest, path to W must be at least as long (or else we would have picked W as the next vertex) So the path through W to V cannot be any shorter! 0

21 orrectness: Inside the loud Prove by induction on # of nodes in the cloud: Initial cloud is just the source with shortest path 0 ssume: Everything inside the cloud has the correct shortest path Inductive step: Only when we prove the shortest path to some node v (which is not in the cloud) is correct, we add it to the cloud When does Dijkstra s algorithm not work?

22 The Trouble with Negative Weight ycles 0-5 E D What s the shortest path from to E? Problem?

23 t each step: Dijkstra s vs FS ) Pick closest unknown vertex ) dd it to finished vertices ) Update distances t each step: ) Pick vertex from queue ) dd it to visited vertices ) Update queue with neighbors Dijkstra s lgorithm readth-first Search Some Similarities:

24 Single-Source Shortest Path Given a graph G = (V, E) and a single distinguished vertex s, find the shortest weighted path from s to every other vertex in G. ll-pairs Shortest Path: Find the shortest paths between all pairs of vertices in the graph. How?

25 nalysis Total running time for Dijkstra s: O( V log V + E log V ) (heaps) What if we want to find the shortest path from each point to LL other points? 5

26 Dynamic Programming lgorithmic technique that systematically records the answers to sub-problems in a table and re-uses those recorded results (rather than re-computing them). Simple Example: alculating the Nth Fibonacci number. Fib(N) = Fib(N-) + Fib(N-) 6

27 Floyd-Warshall for (int k = ; k =< V; k++) for (int i = ; i =< V; i++) for (int j = ; j =< V; j++) if ( ( M[i][k]+ M[k][j] ) < M[i][j] ) M[i][j] = M[i][k]+ M[k][j] Invariant: fter the kth iteration, the matrix includes the shortest paths for all pairs of vertices (i,j) containing only vertices..k as intermediate vertices 7

28 Initial state of the matrix: - a b - c a b c d e d e a b c d e M[i][j] = min(m[i][j], M[i][k]+ M[k][j]) 8

29 Floyd-Warshall - for ll-pairs shortest path a b c d e - a d b e - c a b c Final Matrix ontents d e